Microscopic evidence for anisotropic multigap superconductivity in the CsV₃Sb₅ kagome superconductor

Abstract

The recently discovered kagome superconductor CsV₃Sb₅ (T_c ≃ 2.5 K) has been found to host charge order as well as a non-trivial band topology, encompassing multiple Dirac points and probable surface states. Such a complex and phenomenologically rich system is, therefore, an ideal playground for observing unusual electronic phases. Here, we report anisotropic superconducting properties of CsV₃Sb₅ by means of transverse-field muon spin rotation (μSR) experiments. The fits of temperature dependences of in-plane and out-of-plane components of the magnetic penetration depth suggest that the superconducting order parameter may have a two-gap (s + s)-wave symmetry. The multiband nature of superconductivity could be further supported by the different temperature dependences of the anisotropic magnetic penetration depth γ_λ(T) and upper critical field ${\gamma }_{{{{{\rm{B}}}}}_{{{{\rm{c}}}}2}}(T)$. The relaxation rates obtained from zero field μSR experiments do not show noticeable change across the superconducting transition, indicating that superconductivity does not break time reversal symmetry.

INTRODUCTION

The kagome lattice materials, consisting of a two-dimensional lattice of corner-sharing triangles, have drawn considerable attention in recent years^1,2,3,4. Their electronic structure is characterized by a dispersionless flat band, whose origin lies in the innate kinetic frustration of the kagome geometry, and a pair of Dirac points⁵. Such flat bands, with a high density of electronic states, are generally perceived to quench the kinetic energy and to induce correlated electronic phases when found close to the Fermi level^6,7, as illustrated by the recently discovered superconducting twisted bilayer graphene⁵. The inherent geometrical frustration of kagome systems can be employed to carefully tune their properties, thus aiding in the search of superconductors (SC) with non-phonon mediated pairing mechanisms⁸. A recent example of a kagome superconductor with unconventional coupling is LaRu₃Si₂⁸. Here, the correlation effects from the kagome flat band, the van Hove points on the kagome lattice, and the high density of states from the narrow electronic bands were proposed as key factors for achieving a relatively high transition temperature T_c ≃ 7 K.

Following the recent discovery of the AV₃Sb₅ (A = K, Rb, Cs) family of kagome materials⁹, a slew of interesting and exotic effects have been observed: giant anomalous Hall conductivity^10,11,12, magneto-quantum oscillations^10,13, topological charge order^{14,15,16,17,18,19}, orbital order²⁰, and superconductivity^21,22,23,24. Featuring a kagome network of vanadium atoms interwoven with a simple hexagonal antimony net, the normal state of CsV₃Sb₅ was described as a nonmagnetic Z₂ topological metal^21,22. Furthermore, the observation of CDW order in the normal state of all members of AV₃Sb₅ kagome family has generated significant theoretical and experimental interest. Namely, topological chiral charge order has been reported in AV₃Sb₅ (A = K, Rb, or Cs)^14,16. In KV₃Sb₅, direct evidence for time-reversal symmetry breaking by the charge order was demonstrated using muon spin rotation²⁵.

Regarding superconductivity, a strong diversity in the SC gap symmetry is reported in AV₃Sb₅ family. Proximity-induced spin-triplet pairing was suggested for K_1−xV₃Sb₅²⁶. For CsV₃Sb₅, in particular, there are significant differences in concluding the superconducting gap structure. For instance, multiband superconductivity with sign-preserving order parameter was reported by means of scanning tunneling microscopy (STM) measurements at ultralow temperature²⁷, and magnetic penetration depth measurements using tunnel diode oscillator techniques²⁸. Contrarily, a nodal type superconducting gap symmetry is proposed through thermal conductivity measurements²⁹. Finally, reentrant superconductivity and double SC domes were found under pressure^30,31,32. From a theoretical perspective, several scenarios for electronically mediated, unconventional superconductivity have been discussed³³. The AV₃Sb₅ electronic bands exhibit van Hove singularities close to the Fermi energy—an electronic structural motif shared with other systems, such as the cuprate superconductors or Sr₂RuO₄. A particular feature of the kagome lattice, however, is a sublattice interference mechanism³⁴, by which the Bloch states near each van Hove point are supported on a distinct sublattice. This promotes the relevance of long-range interactions and unconventional pairing states.

To explore unconventional aspects of superconductivity in CsV₃Sb₅, it is critical to measure the superconducting order parameter on the microscopic level through measurements of the bulk properties. Thus, we focus on muon spin rotation/relaxation (μSR) measurements of the magnetic penetration depth³⁵ λ in CsV₃Sb₅. λ is one of the fundamental parameters of a superconductor, since it is related to the superfluid density n_s via:

$$1/{\lambda }^{2}={\mu }_{0}{e}^{2}{n}_{s}/{m}^{* },$$

(1)

where m^* is the effective mass. Most importantly, the temperature dependence of λ is particularly sensitive to the topology of the SC gap: while in a fully gapped superconductor, ${{\Delta }}{\lambda }^{-2}\left(T\right)\equiv {\lambda }^{-2}\left(0\right)-{\lambda }^{-2}\left(T\right)$ vanishes exponentially at low T, whereas in the case of a clean system with line nodes it shows a linear T-dependence.

RESULTS

Sample characterization

To determine sample purity, X-ray diffraction experiments were performed on flux-grown crystals. The powder diffraction pattern of ground crystals can be well fitted using the structure of CsV₃Sb₅ and the fitted lattice parameters are a = 5.50552(2) Å and c = 9.32865(3) Å, close to the previous results²². Additionally, to test the single crystallinity of the samples and determine the orientation for the μSR experiments, X-ray Laue diffraction was performed on the single crystal shown in Fig. 1a, whose diffraction pattern is displayed in Fig. 1c. The crystal was easily aligned; the hexagonal symmetry of the ab-plane is clearly visible from the single crystal, and the crystals grow with the c-axis aligned along the thin direction of the crystal. The diffraction pattern collected in Fig. 1c was analyzed with the OrientExpress program³⁶ and the orientation was confirmed to be along the crystallographic c-axis.

Fig. 1: (Color online) Characterization of CsV₃Sb₅ single crystals.

a Optical microscope photograph of a single crystal with the crystallographic axes highlighted by arrows. Notice the obvious hexagonal symmetry. b Laue x-ray diffraction image of the single crystal along the 001 direction. c X-ray diffraction pattern obtained from a single crystal sample. d Temperature dependence of magnetic susceptibility in an applied field of 1 mT along the ab plane highlighting the superconducting transition. e Temperature dependence of magnetic susceptibility in an applied field of 1 T along the c axis emphasizing the CDW transition at 95 K. f Electrical resistivity as a function of temperature under zero applied field.

The superconductivity of the samples was confirmed by magnetization [Fig. 1d] and resistivity [Fig. 1f] experiments, which show a diamagnetic shift in the sample concurrent with the onset of zero-resistivity at T_c ≃ 2.7 K which is slightly higher than the compared to the T_c (= 2.5 K) value obtained from magnetization measurements. This is most likely due to very tiny filamentary superconducting channels. The onset of charge order is visible in the magnetization measurements in Fig. 1e, corresponding to the anomaly at T_co ≈ 95 K. There is also a slight change in slope visible in the resistivity data presented in Fig. 1f, which occurs at the onset of the charge-ordered state.

Anisotropy in magnetic penetration depth and superconducting gap structure

Two sets of TF-μSR experiments were carried out in the field-cooled state, with the external magnetic field applied parallel to the c − axis, and parallel to the ab (kagome) plane. In both cases the muon spin was perpendicular to the applied field. Note that, for an applied field parallel to the c-axis, the screening currents around the flux-line cores flow in the ab-plane. This allows us to determine the so-called in-plane component of the magnetic penetration depth λ_ab. The TF-μSR time-spectra collected with an external field B_ext = 10 mT applied parallel to the c-axis above (T = 5 K) and well below (T ≃ 0.27 K) the superconducting transition temperature T_c ≃ 2.5 K are shown in Fig. 2a. The corresponding Fourier transforms of the TF-μSR data, representing the magnetic field distribution P(B), are shown in Fig. 2b. The insets in Fig. 2a and Fig. 2b, respectively, represent the geometry of the experiment and the schematic distribution of the magnetic fields within the isotropic flux-line lattice (FLL) with the two components of the magnetic penetration depth, namely λ_a and λ_b being equal: λ_a = λ_b = λ_ab. The TF-μSR time-spectra and the corresponding Fourier transforms collected with B_ext = 10 mT applied along the kagome plane are presented in Fig. 2c and d, respectively. With the field applied along the ab-plane, the screening currents around the vortex cores flow along the ab-plane and c-axis, thus implying that in a set of experiments with B∥ab, λ_ab,c can be determined. Note that, due to the anisotropy, λ_c is longer than λ_ab, which leads to an elongation of the vortex lattice along the c direction [see inset in Fig. 2d].

Fig. 2: Results of TF-μSR experiments.

a and c TF-μSR spectra collected above and below T_c for CsV₃Sb₅ with B_ext∣∣c and B_ext∣∣ab, respectively. The fast damping of the signal in the superconducting state reflects the inhomogeneous field distribution sensed by the muon ensemble as a result of vortex formation. Solid lines through the data points are fits using Eq. (5) in the Suppl. Inset: Scheme of the setup for μSR experiments on a single crystalline CsV₃Sb₅ sample. b and d show the Fourier spectra obtained by fast Fourier transformation of the spectra in a and c, respectively. The inset shows the spatial field variation within a flux line lattice of an anisotropic superconductor with an anisotropy parameter γ_λ ≃ 3, as explained in detail in Refs. ^72,75. The cutoff at low fields corresponds to the minimum in P(B) occurring at the midpoint of three adjacent vortices (${B}_{\min }$). The peak in P(B) arises from the saddle point midway between two adjacent vortices (B_saddle), whereas the long tail towards high fields is due to the region around the vortex cores (${B}_{\max }$). e Temperature variation of the square-root of the second moment of field distribution σ(T) [$\sigma ={M}_{2}^{1/2}$], as measured for the two applied field directions. The horizontal lines denote the relaxation due to nuclear magnetic moments along the two directions. f Internal magnetic field as a function of temperature in the superconducting state. A clear diamagnetic shift, an immanent feature of type-II superconductors, can be seen. g Temperature variation of the skewness parameter, ${\alpha }_{{{{\rm{sk}}}}}={M}_{3}^{1/3}/{M}_{2}^{1/2}$, for the B_ext∣∣c and B_ext∣∣ab set of experiments. The error bars represent the SD of the fit parameters.

The formation of the flux-line lattice (FLL) in the superconductor leads to a nonuniform magnetic field distribution between the vortices [see insets in Fig. 2b and d]. The strong damping of the TF-μSR time-spectra [Fig. 2a and c] and the corresponding broadening of the Fourier transform [Fig. 2b and d] represent exactly this effect. Note that the measured distribution of the magnetic fields in the superconducting state becomes asymmetric, as expected for a well-arranged FLL. All the characteristic features, as e.g., the cutoff at low fields (${B}_{\min }$), the peak due the saddle point between two adjacent vortices (B_sad), and the long tail towards high fields, related to the regions around the vortex core (${B}_{\max }$), are clearly visible for B_ext∣∣c. However. the asymmetric shape is not observed in case of B_ext∣∣ab for two reasons: (i) Long λ_c: long λ leads to smaller field variation within the vortex lattice. (ii) High value of the nuclear moment contribution σ_nm: since the asymmetric line shape caused by the formation of the vortex lattice needs to be convoluted with σ_nm³⁷, which effectively leads to a smearing out the characteristic features of the vortex lattice. The locations of ${B}_{\min }$, B_sad, and ${B}_{\max }$ are shown in the contour plot in the inset of Fig. 2b. To account for the field distribution P(B), the time-domain spectra were analyzed using a skewed Gaussian (SKG) function, which represents the simplest distribution accounting for the asymmetric lineshape (see Suppl. Mater. for a detailed description of the function).

The parameters obtained from the fits are presented in Fig. 2e–g. Figure 2e shows the temperature dependence of the square root of the second moment M₂ (see Suppl. Mater. for detailed calculations) which corresponds to the total depolarization rate σ for two field orientations. Below T_c, the relaxation rate σ starts to increase from its normal-state value due to the formation of the FLL and saturates at low temperatures. The normal-state muon depolarization rate is mostly due to the nuclear magnetic moments and, for CsV₃Sb₅, it has different values for the two field orientations, ${\sigma }_{{{{\rm{nm}}}}}^{| | c}$ = 0.165(3) μs⁻¹ and ${\sigma }_{{{{\rm{nm}}}}}^{| | ab}$ = 0.287(3) μs⁻¹ [shown by horizontal lines in Fig. 2e]. As shown in Fig. 2f, for both field orientations the first moment—which represents the internal field (B_int)—shows a clear diamagnetic shift below T_c, as expected for a type-II superconductor. Note that the T_c value estimated from μSR experiments agrees with that determined from magnetization measurement, so that ${T}_{{{{\rm{c}}}}}^{\mu {{{\rm{SR}}}}}={T}_{{{{\rm{c}}}}}^{{{{\rm{\chi }}}}}\simeq 2.5$ K.

The asymmetric line shape of the field distribution P(B) is characterized by the third moment (M₃) of the field distribution (see Suppl. Mater. for the calculations) and has three characteristic fields: ${B}_{\min }$, ${B}_{\max }$, and B_sad. More accurately, the asymmetry of the line shape is described by its skewness parameter ${\alpha }_{{{{\rm{sk}}}}}=({M}_{3}^{1/3}/{M}_{2}^{1/2})$, which assumes a value of 1.2 for a perfectly arranged triangular vortex lattice^38,39. Distortions or even melting of the vortex lattice structure, which may be caused by variations of temperature or magnetic field, are strongly reflected in α_sk^40,41,42.

Figure 2 (g) shows the temperature evolution of ${\alpha }_{{{{\rm{sk}}}}}^{| | c}$ (for B_ext∣∣c) and ${\alpha }_{{{{\rm{sk}}}}}^{| | ab}$ (for B_ext∣∣ab) for the kagome superconductor CsV₃Sb₅. Notably, in both directions, α_sk(T) remains independent of temperature for T ≲ T_c, with a constant value of ≃ 0.8 for ${\alpha }_{{{{\rm{sk}}}}}^{| | ab}$ and 0.9 for ${\alpha }_{{{{\rm{sk}}}}}^{| | c}$, respectively. We note that near the superconducting transition temperature the μSR response could be well fitted by the single Gaussian line (i.e., the reduced χ² of SKG and single Gaussian fits become almost equal). Since for the symmetric P(B) distribution α_sk stays exactly at zero, this leads to the sudden change of α_sk at T ~ 2.2 K, i.e., ≃ 0.3 K below T_c. This observation suggests that very close to T_c, the FLL is slightly distorted but does not disturb the determination of the temperature evolution of the superfluid density along different crystallographic directions (as we show later), which is the main goal of the present study.

We estimate the superconducting contribution to the depolarization rate, σ_sc, by quadratically subtracting the temperature-independent nuclear magnetic moment contribution σ_nm (obtained above T_c) from the the total depolarization rate σ (i.e., ${\sigma }_{{{{\rm{sc}}}}}^{2}={\sigma }^{2}-{\sigma }_{{{{\rm{nm}}}}}^{2}$).

σ_sc can be expressed as a function of the reduced field b = $\frac{B}{{B}_{{{{\rm{c}}}}2}}$ (B_c2 being the upper critical field) and the Ginzburg-Landau coefficient κ by the relation developed by Brandt^43,44

$${\sigma }_{{{{\rm{sc}}}}}[\mu {{{{\rm{s}}}}}^{-1}]\approx 4.83\times 1{0}^{4}(1-b)[1+1.21{(1-\sqrt{b})}^{3}]{\lambda }^{-2}[{{{{\rm{nm}}}}}^{-2}],$$

(2)

where λ is the magnetic penetration depth. Thus, we obtain the temperature dependence of ${\lambda }_{{{{\rm{ab}}}}}^{-2}$ (for B_ext∥c) and ${\lambda }_{{{{\rm{ab,c}}}}}^{-2}$ (for B_ext∥ab). In the case of an anisotropic superconductor, the magnetic penetration depth is also anisotropic. In the present case, by considering an anisotropic effective mass tensor, ${\lambda }_{{{{\rm{ab}}}}}^{-2}$ and ${\lambda }_{{{{\rm{ab,c}}}}}^{-2}$, ${\lambda }_{{{{\rm{c}}}}}^{-2}$ can be estimated (see Suppl. Mater. for the detailed analysis). In this way, we can directly compare the anisotropy of the magnetic penetration depth. Figure 3a and b shows the temperature evolution of ${\lambda }_{{{{\rm{ab}}}}}^{-2}$ and ${\lambda }_{{{{\rm{c}}}}}^{-2}$, respectively.

Fig. 3: Temperature dependence of London penetration depth and anisotropy parameter.

a and b Temperature variations of the in-plane and out-of-plane component of the inverse squared magnetic penetration depths, ${\lambda }_{ab}^{-2}(T)$ and ${\lambda }_{c}^{-2}(T)$. Data are fitted with a two gap (s + s)-wave, and a nodal d-wave model, as described by the phenomenological α-model. c Magnetic penetration depth anisotropy (γ_λ) and upper critical field anisotropy (${\gamma }_{{B}_{{{{\rm{c}}}}2}}$) as a function of temperature. ${\gamma }_{{B}_{{{{\rm{c}}}}2}}(T)$ is obtained by considering the SC transition temperature T_c as the temperature where resistivity approaches zero. The error bars represent the SD of the fit parameters.

As mentioned in the introduction, there are only a few studies involving different experimental techniques to address the superconducting gap structure of the CsV₃Sb₅ kagome superconductor. However, there is no consensus among them. To determine whether the superconducting gap structure of this compound is of single-gap, multigap, or even of nodal nature, we analyzed the temperature dependence of magnetic penetration depth. We analyzed λ(T) data using the following expression:

$$\frac{{\lambda }^{-2}(T,{{{\Delta }}}_{0})}{{\lambda }^{-2}(0,{{{\Delta }}}_{0})}=1+\frac{1}{\pi }\int\nolimits_{0}^{2\pi }\,\,\int\nolimits_{{{\Delta }}(T,\phi )}^{\infty }\frac{\partial f}{\partial E}\frac{E\,{{{\rm{d}}}}E\,{{{\rm{d}}}}\phi }{\sqrt{{E}^{2}-{{\Delta }}{(T,\phi )}^{2}}},$$

(3)

where $f={(1+E/{k}_{{{{\rm{B}}}}}T)}^{-1}$ represents the Fermi-distribution function. The temperature- and angle-dependent gap function is described by Δ(T, ϕ) = Δ₀δ(T/T_c)g(ϕ), where Δ₀ is the maximum gap value at T = 0, while the temperature dependence of the gap function is^45,46$\delta (T/{T}_{{{{\rm{c}}}}})=\tanh \{1.821[1.018{({T}_{{{{\rm{c}}}}}/T-1)}^{0.51}]\}$. Here, g(φ) corresponds to the angular dependence of the gap and takes a value of 1 for s-wave and $| \cos (2\varphi )|$ for d-wave gap symmetry. Motivated by recent studies reporting a nodal-gap structure in CsV₃Sb₅, evidenced by a non-zero value of the residual linear term of its thermal conductivity at zero field together with its rapid increase with field, we tried to fit the data using a d-wave model²⁹. However, as shown in Fig. 3a and b, a d-wave model cannot describe the data well. On the contrary, recent tunneling experiments²⁷, tunnel-diode oscillator (TDO) based results, along with specific heat measurements conjointly indicate a multiband nature of superconductivity in CsV₃Sb₅²⁸. Thus, we proceeded to fit the ${\lambda }_{{{{\rm{c}}}}}^{-2}(T)$ and ${\lambda }_{{{{\rm{ab}}}}}^{-2}(T)$ data simultaneously with a two-gap scenario using a weighted sum:

$$\frac{{\lambda }^{-2}(T)}{{\lambda }^{-2}(0)}=x\frac{{\lambda }^{-2}(T,{{{\Delta }}}_{0,1})}{{\lambda }^{-2}(0,{{{\Delta }}}_{0,1})}+(1-x)\frac{{\lambda }^{-2}(T,{{{\Delta }}}_{0,2})}{{\lambda }^{-2}(0,{{{\Delta }}}_{0,2})}.$$

(4)

Here x is the weight associated with the larger gap and Δ_0,i (i = 1, 2 are the band indices) are the gaps related to the first and second band.

The fit of λ_ab(T) and λ_c(T) with a two-gap (s + s)-wave model was performed by assuming similar gap values [Δ_1(2),ab = Δ_1(2),c], but different weighting factors for the two directions (x_ab ≠ x_c). A fit with a d-wave model was also performed by considering a similar gap value along two different directions. It is evident that the two-gap model follows the experimental data very well [Fig. 3a and b]. Table 1 summarizes the different superconducting parameters as obtained from the fits. Note that the TF-μSR measurements are well described by gap values which are very close to those obtained from tunneling measurements (Δ₁ = 0.57 meV, Δ₂ = 0.3 meV, Δ₃ = 0.45 meV)²⁷. On the other hand, as shown in Fig. 3a and b, a d-wave model does not describe the data well. This conclusion could be reinforced also by the systematically larger value of the reduced ${\chi }_{r}^{2}$ in case of a d-wave model (1.32) compared to the s + s-wave model fit (1.20). In this regard, it is worthwhile mentioning that recent NMR studies on CsV₃Sb₅ clearly indicate a coherence peak below T_c⁴⁷, further supporting nodeless superconductivity in CsV₃Sb₅. Although thermal conductivity measurements at ultralow temperatures²⁹ suggest the presence of nodes in the superconducting energy gap, our data do not seem to be consistent with with nodal superconductivity, a result further confirmed by NMR/NQR⁴⁷, tunneling²⁷, specific heat, and magnetic penetration depth based on TDO measurements²⁸. Further studies are needed in order to clarify the discrepancy between the macroscopic and local-probe techniques.

Table 1 Superconducting parameters determined from fits to the temperature dependence of λ(T) derived from TF-μSR experiments, using an (s + s)-wave and a d-wave model. Here ${\chi }_{r}^{2}$ represents the reduced χ² values for different fit models.

Furthermore, we determined the anisotropy of the magnetic penetration depth, γ_λ, defined as:

$${\gamma }_{\lambda }=\frac{{\lambda }_{{{{\rm{c}}}}}}{{\lambda }_{{{{\rm{ab}}}}}}=\frac{{\lambda }_{{{{\rm{ab}}}}}^{-2}}{{\lambda }_{{{{\rm{ab,c}}}}}^{-2}}.$$

(5)

By using the ${\lambda }_{{{{\rm{ab}}}}}^{-2}(T)$ and ${\lambda }_{{{{\rm{c}}}}}^{-2}(T)$ values, we obtain the temperature dependence of the magnetic penetration depth anisotropy as presented in Fig. 3c. According to the phenomenological Ginzburg-Landau theory for uniaxial anisotropic superconductors, the various anisotropies, such as the magnetic penetration depth anisotropy γ_λ and the upper critical field anisotropy ${\gamma }_{{B}_{{{{\rm{c}}}}2}}$, can be accounted for by a common parameter:^48,49

$${\gamma }_{\lambda }=\frac{{\lambda }_{{{{\rm{c}}}}}}{{\lambda }_{{{{\rm{ab}}}}}}=\sqrt{\frac{{m}_{c}^{* }}{{m}_{{{{\rm{ab}}}}}^{* }}}={\gamma }_{{B}_{{{{\rm{c}}}}2}}=\frac{{B}_{{{{\rm{c}}}}2}^{\parallel {{{\rm{a}}}}b}}{{B}_{{{{\rm{c}}}}2}^{\parallel {{{\rm{c}}}}}}=\frac{{\xi }_{{{{\rm{ab}}}}}}{{\xi }_{{{{\rm{c}}}}}}.$$

(6)

In the above equation, ξ represents the coherence length. In order to compare γ_λ and ${\gamma }_{{B}_{{{{\rm{c}}}}2}}$, we analyzed the electrical transport data in the presence of various applied fields and estimated the temperature dependence of the upper critical field anisotropy ${\gamma }_{{B}_{{{{\rm{c}}}}2}}(T)$ = B_c2,ab(T)/B_c2,c(T), where B_c2,ab(T) and B_c2,c(T) are the upper critical fields corresponding to zero values of resistivity for B∥ab and B∥c (see Supplementary Fig. 1). Figure 3c depicts also the temperature dependence of ${\gamma }_{{B}_{{{{\rm{c}}}}2}}$. Interestingly, γ_λ(T) changes slightly from γ_λ ≃ 1.6 close to T_c to γ_λ ≃ 3.3 close to T = 0 K. Conversely, ${\gamma }_{{B}_{{{{\rm{c}}}}2}}(T)$ varies strongly from ${\gamma }_{{B}_{{{{\rm{c}}}}2}}\simeq 7.1$ close to T_c to ${\gamma }_{{B}_{{{{\rm{c}}}}2}}\simeq 8.5$ when T_c ≃ 0.3 K.

ZF-μSR experiment in the superconducting state

Finally, to check whether superconductivity breaks time-reversal symmetry in CsV₃Sb₅, we performed zero-field (ZF) μSR measurements at different temperatures across T_c. A time-reversal symmetry-breaking superconducting state is one of the hallmarks of certain unconventional superconductors; in particular, in many Re-based superconductors⁵⁰, Sr₂RuO₄^51,52,53, Ba_1−xK_xFe₂As₂⁵⁴, LaNiC₂⁵⁵, La₇Ir₃⁵⁶ etc. Due to the high muon sensitivity to low magnetic fields, μSR has been the technique of choice for studying it. Here, the ZF-μSR spectra are well described by a damped Gaussian Kubo-Toyabe depolarization function⁵⁷ (see Supp. Mater. for a detailed description of analysis), which considers the field distribution at the muon site created by both nuclear and electronic moments. The (ZF) μSR experiments were carried out with the muon spin rotated by ~43^∘ with respect to the direction of muon momentum. This configuration allows us to use two different detector pairs, namely, Forward(F)- Backward(B) and Left(L)-Right(R), to probe the anisotropic ZF response (see Supplementary Fig. 2 for a schematic view of the experimental geometry). Figure 4a and b shows the ZF-μSR spectra collected above and below T_c. Figure 4c shows the temperature dependence of the electronic relaxation rates for different detectors. The shaded regions represent the statistical scattering of data points. It is evident from Fig. 4c that the relaxation rates do not show any noticeable increase below T_c, indicating that superconductivity does not break TRS in CsV₃Sb₅ within the statistical accuracy, similar to the recent results on the related system KV₃Sb₅²⁵. Further ZF experiments with high statistics would be interesting in order to confirm or disprove the presence of time reversal symmetry breaking in the superconducting ground state.

Fig. 4: Results of ZF-μSR experiments.

a and b ZF-μSR spectra above and below T_c collected in zero field with the initial muon spin polarization being S_μ(0)∣∣ab and S_μ(0)∣∣c, respectively. The solid lines through the spectra are fits as described in the text. c The resulting ZF exponential relaxation rates Λ(T) are almost independent of temperature, thus indicating a preserved TRS. The error bars represent the SD of the fit parameters.

DISCUSSION

The probability field distribution determined experimentally shows a highly asymmetric lineshape, indicative of a well-ordered FLL in the vortex state of the superconductor CsV₃Sb₅, and it remains almost independent of temperature until very close to T_c. This observation unambiguously suggests that the FLL in CsV₃Sb₅ is well arranged in the superconducting state and it gets slightly distorted only in the vicinity of T_c. In general, the change of α_sk as a function of magnetic field and temperature is associated with the vortex lattice melting^40,41,58, and/or a dimensional crossover from a three-dimensional (3D) to a two-dimensional (2D) type of FLL^39,41. Both processes are thermally activated and caused by increased vortex mobility via a loosening of the inter- or intraplanar FLL correlations³⁹. Another possibility involves the rearrangement of the vortex lattice induced by a change of the anisotropy coefficient γ_λ = λ_c/λ_ab⁵⁹. Since CsV₃Sb₅ has a very small superconducting anisotropy (γ_λ ≃ 3, see below), we can rule out a possible vortex-melting scenario. As for the anisotropy-induced FLL rearrangement, the temperature evolution of α_sk measured in B_ext∥ab and B_ext∥c experiments are expected to be very much different⁵⁹, which is also not the case here [see Fig. 3c]. Therefore, we are left with the explanation that close to T_c, where the broadening of μSR signal caused by formation of FLL becomes comparable with or even smaller than the relaxation caused by the nuclear magnetic moments [straight lines in Fig. 2e], the shape of P(B) distribution is dominated by the symmetric ‘nuclear’ term, which effectively pushes α_sk to zero shortly before the superconducting transition temperature T_c is reached.

Furthermore, a detailed analysis of the λ(T) data reveals the presence of two superconducting gaps at the Fermi surface, with gap values of 0.6 and 0.23 meV. This conclusion is in agreement with recent reports involving different experimental techniques^28,60. As μSR is a bulk probe, we conclude that the bulk superconducting gap of this compound consists of two s-wave gaps rather than a nodal gap. Another interesting observation is the fact that the T_c/λ⁻²(0) ratio for CsV₃Sb₅ in both field orientations is comparable to those of high-temperature unconventional superconductors and iron-pnictides^61,62. Systems with a small T_c/λ⁻²(0) ~ 0.00025–0.015 are usually considered to be BCS-like, while large T_c/λ⁻²(0) values are expected only in the BEC-like picture and is considered a hallmark feature of unconventional superconductivity. This approach has become a key feature to characterize BCS-like (so-called conventional) and BEC-like superconductors. Remarkably, in CsV₃Sb₅, T_c/λ⁻²(0) is as high as ~ 0.2 (for ${\lambda }_{{{{\rm{ab}}}}}^{-2}(0)$) - 2.2 (${\lambda }_{{{{\rm{c}}}}}^{-2}(0)$), where the lower limit is comparable to the unconventional transition metal dichalcogenide superconductors⁶³ and the upper limit is close to T_c/λ⁻²(0) ~ 4 of hole-doped cuprates^61,62,64. This point towards an unconventional pairing mechanism in the kagome superconductor CsV₃Sb₅.

Moreover, we observe a clear difference in the temperature dependence of the anisotropies related to magnetic penetration depth and upper critical fields, again signaling a clear deviation from the Ginzburg-Landau theory. This situation finds a clear parallel with the data from the well-known two-gap superconductor MgB₂^65,66, Fe-based Sm- and Nd-1111 systems^67,68, 122 pnictide superconductors: Ba(Fe_1−xCo_x)₂As₂^69,70, (Ba_1−xK_x)Fe₂As₂⁵⁹; FeSe_0.5Te_0.5⁷¹, CaKFe₄As₄⁷², etc., where two different temperature variations of γ_λ(T) and ${\gamma }_{{B}_{{{{\rm{c}}}}2}}(T)$ were attributed to multiband superconductivity. Thus, in comparison to well-established multigap superconductors, we find further support for the multigap behavior in this compound. Therefore, our results provide microscopic evidence of anisotropic multigap superconductivity in the kagome superconductor CsV₃Sb₅ and encourage further theoretical and experimental research on the kagome superconductors.

Finally, ZF-μSR experiments suggest that across T_c, the relaxation rates do not change. This suggests that within the statistical accuracy, no clear signature of TRS breaking is observed in the superconducting state of CsV₃Sb₅.

METHODS

Sample preparation

Single crystals of CsV₃Sb₅ were grown from Cs ingots (purity 99.9%), V 3-N powder (purity 99.9%) and Sb grains (purity 99.999%) using the self-flux method, similar to the growth of RbV₃Sb₅⁷³. The eutectic mixture of CsSb and CsSb₂ was mixed with VSb₂ to form a composition with approximately 50 at% Cs_xSb_y and 50 at% VSb₂. The mixture was put into an alumina crucible and sealed in a quartz ampoule under partial argon atmosphere. The sealed quartz ampoule was heated to 1273 K in 12 h and kept there for 24 h. Then it was cooled down to 1173 K at 50 K/h and further to 923 K at a slower rate. Finally, the ampoule was removed from the furnace and decanted with a centrifuge to separate the CsV₃Sb₅ crystals from the flux. The obtained crystals have a typical size of 4 × 4 × 1 mm³ and are stable in air over a period of at least several months. As shown in Fig. 1a and b, the flux-grown single crystals possess an obvious hexagonal symmetry, while the X-ray Laue diffraction images demonstrate the single crystallinity of the material. The XRD pattern was collected using a Bruker D8 x-ray diffractometer with Cu K_α radiation (λ = 0.15418 nm) at room temperature.

Magnetic susceptibility and electrical transport measurements

The magnetization measurements were performed in a Quantum Design magnetic property measurement system SQUID magnetometer under field-cooled and zero-field-cooled conditions. Electrical transport measurements were carried out in a Quantum Design physical property measurement system (PPMS-14T). The longitudinal electrical resistivity was measured using a four-probe method with the current flowing in the ab plane. For measurements in the temperature range 0.3–2 K we employed a Heliox recondensing He-3 system, which is equipped with a superconducting NbTi twisted pair cable for the measurements.

μSR-experiments

We performed transverse field (TF) and zero field μSR experiments using the Dolly spectrometer (πE1 beamline) at the Paul Scherrer Institute (Villigen, Switzerland). Since the crystals were rather thick (~1 mm), they were mounted in a single layer using Apiezon N grease, to form a mosaic covering an area of 7 × 7 mm². The Dolly spectrometer is equipped with a standard veto setup, providing a low-background μSR signal. All TF- μSR experiments were done after field-cooling the sample with the applied field either along the kagome plane or perpendicular to it. The μSR time spectra were analyzed using the open software package MUSRFIT⁷⁴.

Notes added

While preparing this manuscript, we came to know about another μSR work studying the high temperature charge ordered phase of CsV₃Sb₅ [arXiv:2107.10714 (2021) (https://arxiv.org/abs/2107.10714)].